Optimal. Leaf size=150 \[ \frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10240000 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52,
56, 222} \begin {gather*} \frac {1269748711 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10240000 \sqrt {10}}-\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}-\frac {7 (1-2 x)^{3/2} (2256 x+3821) (5 x+3)^{5/2}}{32000}-\frac {953981 (1-2 x)^{3/2} (5 x+3)^{3/2}}{384000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {5 x+3}}{1024000}+\frac {115431701 \sqrt {1-2 x} \sqrt {5 x+3}}{10240000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{60} \int \left (-315-\frac {987 x}{2}\right ) \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {953981 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{64000}\\ &=-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {10493791 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{256000}\\ &=-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {115431701 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2048000}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{20480000}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10240000 \sqrt {5}}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10240000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 88, normalized size = 0.59 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-1451592441-2354035875 x+4157008580 x^2+14549698400 x^3+19494864000 x^4+12890880000 x^5+3456000000 x^6\right )-3809246133 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{307200000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 138, normalized size = 0.92
method | result | size |
risch | \(-\frac {\left (691200000 x^{5}+2163456000 x^{4}+2600899200 x^{3}+1349400160 x^{2}+21761620 x -483864147\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{30720000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1269748711 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{204800000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (13824000000 x^{5} \sqrt {-10 x^{2}-x +3}+43269120000 x^{4} \sqrt {-10 x^{2}-x +3}+52017984000 x^{3} \sqrt {-10 x^{2}-x +3}+26988003200 x^{2} \sqrt {-10 x^{2}-x +3}+3809246133 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+435232400 x \sqrt {-10 x^{2}-x +3}-9677282940 \sqrt {-10 x^{2}-x +3}\right )}{614400000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 104, normalized size = 0.69 \begin {gather*} -\frac {9}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {2727}{400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {270711}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2147273}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {10493791}{512000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1269748711}{204800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {10493791}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 82, normalized size = 0.55 \begin {gather*} \frac {1}{30720000} \, {\left (691200000 \, x^{5} + 2163456000 \, x^{4} + 2600899200 \, x^{3} + 1349400160 \, x^{2} + 21761620 \, x - 483864147\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1269748711}{204800000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 71.39, size = 796, normalized size = 5.31 \begin {gather*} - \frac {3773 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{32} + \frac {3283 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} - \frac {1071 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{8} + \frac {621 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{16} - \frac {135 \sqrt {2} \left (\begin {cases} \frac {1771561 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{161051} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}} \left (20 x + 1\right )^{3}}{170069856} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{15488} - \frac {13 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{1024}\right )}{15625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (111) = 222\).
time = 0.58, size = 356, normalized size = 2.37 \begin {gather*} \frac {9}{512000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{4000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {921}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {883}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {141}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {36}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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